Adaptive and efficient fractional-order modeling of nonlinear dynamical systems: Addressing memory and order selection challenges

Tariq Ali; Sana Yasin; Umar Draz; Husam S. Samkari; Mohammad Hijji; Mohammed F. Allehyani; Muhammad Ayaz; Tariq Ali; Sana Yasin; Umar Draz; Husam S. Samkari; Mohammad Hijji; Mohammed F. Allehyani; Muhammad Ayaz

doi:10.3934/math.2026878

AIMS Mathematics

2026, Volume 11, Issue 7: 21704-21726. doi: 10.3934/math.2026878

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Adaptive and efficient fractional-order modeling of nonlinear dynamical systems: Addressing memory and order selection challenges

1.
Artificial Intelligence and Sensing Technology Research Center (AIST), University of Tabuk, Tabuk 71491, Saudi Arabia
2.
Faculty of Computers and Information Technology, University of Tabuk, Tabuk, 71491, Saudi Arabia
3.
School of Intelligent Science and Engineering, Harbin Institute of Technology, 518055, Shenzhen, China and Computer Science Department, University of Okara, Okara, 56300, Pakistan
4.
School of Computer Science and Technology, Harbin Institute of Technology, 518055, Shenzhen, China and Department of Computer Science, University of Sahiwal, Sahiwal, 57000, Pakistan
5.
Department of Electrical Engineering, University of Tabuk, Tabuk, 47713, Saudi Arabia

Received: 16 April 2026 Revised: 24 June 2026 Accepted: 01 July 2026 Published: 22 July 2026
MSC : 34A08, 68T07

The nonlinear dynamical systems exhibit the long-memory characteristics is not an easy task due to their time-dependent and nonlocal property. The integer-order models cannot represent the memory property of the system, whereas the fixed fractional-order models suffer from inefficiency along with high computational cost. To overcome these problems, in this paper, we propose technique, which incorporates the order adjustment with effective memory reduction. The fractional order ($ \alpha $) is adjusted according to the variation in the system states, allowing the representation of adaptive memory, whereas the redundant components in memory are kept minimum in the process. To validate the proposed approach, the performance was compared with that of the and the fixed models using the well-known Lorenz attractor dataset. It observed that our method the baseline approaches in terms of differences of 0.0376, 0.0294, 4.12%, and $ R^2 $ score (0.978), respectively. Moreover, the model better ability in preserving the underlying chaotic dynamics.
- fractional-order modeling,
- dynamical systems,
- fractional order,
- optimization,
- Lorenz
Citation: Tariq Ali, Sana Yasin, Umar Draz, Husam S. Samkari, Mohammad Hijji, Mohammed F. Allehyani, Muhammad Ayaz. Adaptive and efficient fractional-order modeling of nonlinear dynamical systems: Addressing memory and order selection challenges[J]. AIMS Mathematics, 2026, 11(7): 21704-21726. doi: 10.3934/math.2026878

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Abstract

The nonlinear dynamical systems exhibit the long-memory characteristics is not an easy task due to their time-dependent and nonlocal property. The integer-order models cannot represent the memory property of the system, whereas the fixed fractional-order models suffer from inefficiency along with high computational cost. To overcome these problems, in this paper, we propose technique, which incorporates the order adjustment with effective memory reduction. The fractional order ($ \alpha $) is adjusted according to the variation in the system states, allowing the representation of adaptive memory, whereas the redundant components in memory are kept minimum in the process. To validate the proposed approach, the performance was compared with that of the and the fixed models using the well-known Lorenz attractor dataset. It observed that our method the baseline approaches in terms of differences of 0.0376, 0.0294, 4.12%, and $ R^2 $ score (0.978), respectively. Moreover, the model better ability in preserving the underlying chaotic dynamics.

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